As for methods which rely specifically on PDE, the theory started with the paper of Bronsard and Reitich [1]},
who proved local existence and uniqueness provided the initial network is regular and of class \({\mathcal {C}}^{2+\alpha }\) ,
with the sum of the curvatures at each triple junctions equal to zero. This last condition is tantamount to the
first compatibility condition. Their proof is based on a standard linearization procedure and a fixed point argument.
If the initial datum is a regular network of class \({\mathcal {C}}^2\) , but without any restriction on the curvatures at the junctions,
then [2]} proves existence but not uniqueness. Uniqueness was proved in [3]} for initial
networks in \(W^{2-{2}{p}}_p\)  for any \(p\in (3,\infty )\) . A fortiori, this gives uniqueness for regular networks
of class \({\mathcal {C}}^2\) . Noting that \(W^{2-{2}{p}}_p \hookrightarrow {\mathcal {C}}^{1+\alpha }\) , we see that the space
\(W_p^1\left((0,T);L_p((0,1);\mathbb {R}^2)\right)\cap L_p\left((0,T); W_p^2\left((0,1);\mathbb {R}^2\right)\right)\) 
where solutions lie embeds into the space of bounded, uniformly continuous functions from \([0,T]\)  to \(C^{1+\alpha }\) .
This degree of regularity is needed in order to get the contraction estimates in the fixed point argument
and to be able to invoke classical parabolic theory for the associated linear system.
